Averaging operators over nondegenerate quadratic surfaces in finite fields

TL;DR

This paper improves understanding of averaging operators over nondegenerate quadratic surfaces in finite fields, removing previous bounds and solving key problems in even dimensions with specific subspace conditions.

Contribution

It eliminates the logarithmic bounds in averaging estimates over quadratic surfaces in finite fields for even dimensions with certain subspace structures.

Findings

Removed the logarithmic bound in averaging estimates

Settled averaging problems for even dimensions with specific subspaces

Extended the understanding of quadratic surface operators in finite fields

Abstract

We study mapping properties of the averaging operator related to the variety $ V={x\in \mathbb F_q^d: Q(x)=0},$ where $Q(x)$ is a nondegenerate quadratic polynomial over a finite field $\mathbb F_q$ with $q$ elements. This paper is devoted to eliminating the logarithmic bound appearing in the paper of Koh and Shen. As a consequence, we settle down the averaging problems over the quadratic surfaces $V$ in the case when the dimensions $d\geq 4$ are even and $V$ contains a $d/2$-dimensional subspace.